Scott CH Huang COM5336 Cryptography Lecture 11 Euclidean Domains & Division Algorithm Scott CH Huang COM 5336 Cryptography Lecture 10.

Scott CH Huang

COM5336 CryptographyLecture 11

Euclidean Domains & Division Algorithm

Scott CH Huang

COM 5336 Cryptography Lecture 10

Scott CH Huang COM 5336

Groups

• Binary operations on a set is a mapping• A set w/ an operation satisfying

1. Closure2. Associativity3. Identity4. Inverse

• The most fundamental algebraic structure• Semi-groups: 1 & 2 only.• Abelian groups: commutative groups.


Rings

• A set R with two operations: + and *.– +: commutative.– *: not necessarily commutative.

• (R,+) forms an abelian group.• (R,*) forms a semi-group (i.e. no identity and inverse)• Distributivity• Ring v.s. Ring with 1 (mult. identity).


Integral Domains

• Domain = Ring w/o zero-divisors– ab=0 implies a=0 or b=0– One-sided cancellation law

• Integral Domain = Commutative domain w/ 1.– Two-sided cancellation law


Euclidean Domains

• A Euclidean Domain is an integral domain with the notion of size.• The notion of size enables us to apply the Division Algorithm and

therefore Euclid’s Algorithm.• Size of a≠0, denoted by g(a) is a nonnegative integer s.t.

– g(a)≤g(ab), for all b≠0.– For all a,b≠0, there exists q,r s.t. a=qb+r, w/ r=0 or g(r)<g(b)


Division Algorithm

• A theorem in mathematics which precisely expresses the outcome of the usual process of division of integers.

• Its name is a misnomer.• It is not a true algorithm.

– A well-defined procedure for achieving a specific task


Division Rings

• A ring with unit in which division is possible.– i.e. every nonzero element has a multiplicative inverse.

• A division ring is NOT necessarily commutative.– But finite division rings must be commutative (Wedderburn's little theorem).

• A field is a commutative division ring.– Therefore all finite division rings are finite fields.


Relationship of Algebraic Structures

Euclidean Domain

Integral Domain

Ring w/ unit

Ring

Commutative ring w/ unit

Division Ring

Field


Division in a Euclidean Domain

• a|b: ‘a’ divides ‘b’ iff there exists c s.t. b=ac– a,b,c D, a Euclidean domain.

• If a|b1, a|b2,…, then a is a common divisor of b1,b2,…• If d is a common divisor of b1,b2,…, and every common divisor divides d,

then d is a greatest common divisor (GCD) of b1,b2,…• In fact, the concept of GCD can be extended to certain integral domains

called Principal Ideal Domains.


GCD in Algebraic Structures

algebraic structure requirement properties

Integral Domain loose GCD can be defined.

Pricipal Ideal Domain stricter GCD can be defined and exists.

Euclidean Domain strictest GCD can be defined and can be found


GCD may not exists in an Integral Domain

Note that Both d1, d2 are common divisors of b1, b2 ,

so b1, b2 has no greatest common divisors.


GCD exists in a Euclidean Domain

• If , then d can be expressed as a linear combination of a,b.

• If D is a Euclidean domain and , then d can be expressed as a linear combination of a,b

• How to calculate the GCD?


Euclid’s Inspiring Lemma

• gcd(s,t)=gcd(s,t-rs) for all s,t,r in a Euclidean domain D.• This lemma directly results in Euclid’s algorithm.


Euclid’s Algorithm

int gcd(s,t){ while (s!=0){ u=s; s= t mod s; t=u; } return t;}


Theorem #1

Let t be an element in a Euclidean domain Dand m,n be two positive integers. Then

*Hint: (tn-1)-tn-m (tm-1)= tn-m -1


Corollary #1

Let x be an element in a Euclidean domain D and q,n,d be positive integers. Then


Conceptually

Group +, -

Ring +, -, *

Integral Domain +, -, * and “cancellation”

Euclidean Domain +, -, * and “division algorithm”

Field +, -, *, /


Some Examples

• • • • •

Euclidean domainring w/ 1finite fieldcommutative ring w/ 1Euclidean domain


More Examples (cont’d)

• The set of polynomials over an arbitrary field with polynomial addition & multiplication.

• The set of polynomials with two variables x,y over an arbitrary field with polynomial addition & multiplication.


Factorization in Euclidean Domains

• We wish to establish a “Fundamental Theorem of Arithmetic” in Euclidean domains.

• Fundamental Theorem of Arithmetic (aka Unique-Prime-Factorization Theorem)– Any integer greater than 1 can be written as a unique product (up to ordering

of the factors) of prime numbers.• In order to do that, it’s vital to introduce the idea of a “prime number” in

Euclidean domains.


Preliminaries

• Let D be an integral domain. A unit u D is any divisor of 1.– In the integer ring, the units are ±1. In the Gaussian integer ring, ±1, ±i are

units.

• a, b D are associates if a=ub for some unit u.– In the integer ring, +3, -3 are associates. In the Gaussian integer ring, 1+ i, 1- i

are associates.

• A factorization of b is an expression of the form b=a1a2· · · ar. If each of the ai’s are either a unit or an associate of b, this is a trivial factorization.


Irreducible Elements in Integral Domains

• A element p D, an integral domain, is called irreducible iff every factorization of p is trivial.

• We do not consider units to be irreducible.• b D. d|b. If d is not an associate of b, then it is called a proper divisor.• Irreducible elements have no proper divisors other than units.


Primes in Integral Domains

• A nonzero, non-unit element p D, an integral domain, is called prime iff the following property holds.– If p|ab, then either p|a or p|b for a,b D.


Primes vs Irreducible Elements

• In an integral domain, every prime is irreducible.• In a Principal Ideal Domain (PID), every irreducible element is prime.• In our textbook, only Euclidean domains are discussed. The author did not

distinguish between primes and irreducible elements and regarded them as synonyms.


Relative Primality

• In a PID, two elements a,b are relatively prime iff gcd(a,b)=1. (remember that GCD must exists in a PID)

• In a Euclidean domain, if p does not divide a and p is prime, then p and a are relatively prime.


Some Properties

• In a Euclidean domain, if p does not divide a, then there exist s,t such that ps+at=1.

• In a Euclidean domain, if a is a proper divisor of b, then g(a)<g(b).


Unique Factorization

• Theorem 3.6: In a Euclidean domain, if b is not a unit, then b can be factorized as a product of primes:– b=p1p2 · · · pn

– If b can be factorized in another way as b=q1q2 · · · qn , then after appropriate renumbering, pi qi are associates for all i.

• In short, Euclidean domains are Unique Factorization Domains (UFD).


Euclidean Domains, PIDs, UFDs

Euclidean Domain

Integral Domain

Field

*Principal Ideal Domain*

Unique Factorization Domain


Example of a non-UFD

• Consider the integral domain

• are irreducible.•

Scott CH Huang COM5336 Cryptography Lecture 11 Euclidean Domains & Division Algorithm Scott CH Huang COM 5336 Cryptography Lecture 10.

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